4. Let r(s) be a unit speed curve with a curvature of zeroc fo. prove that the image r lies in a plane if and only if 3 there exist a point X, E IR such that every osculating plane passes through X, ·

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Q: 1. Prove that fo r*dr + rydy = where C1 is the line segment from (0,0) to (1,0) by using…

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Q: Prove that all normal to the curve x=acos+atsint,y=asint-atcost.Are at a distance a from the origin

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Q: 04. Show that the Gauas and mean curvatures on, X-(u+v,u-v, uv) at u=1, v=1 are K=1/16 and H= JI.…

A: Given: 1) X=(u+v,u-v,uv) at u=1,v=1. 2) The paraboloid is xy=aZ.

Q: 2. Show that if a : [a, b] → R" is a regular parameterization of a curve then the curvature at α(t)…

A: Let, is a regular parameterization of a curve. To show the curvature at  is .

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A: Curvature of  a graph y= f(x)  k=d2ydx21+dydx232

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Q: 2.Let a be a unit speed curve lying on a sphere center c and radius a > 0. Find the minimum value of…

A: Given That : Let α be the unit speed  curve lying on a sphere center c and radius a>0. To Find :…

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A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 8) If curvature of a space curve at (2,3) is 8 and curvature at (7,1) t=7 is 1, what can be said…

A: ANSWER:

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Q: 3) Prove thet the xy-plane with constant curva tue k= urve In is o circle of radious

A: We have to prove that a curve in the xy plane with constant curvature k=1r is a circle of radius r

Q: a) Find a plane curve whose signed curvature is given by k,(s) where a is a positive constant and s>…

A: The given signed curvature is ks(s)=12as

Q: Find the curvature of the space curve. r (1) = -8i + (1 + 4) j + (8 + In(cos t)) k %3D OK = sec t…

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Q: Show that the regular curve a = a(s) is a straight line if and only if its curvature k is dentically…

A: WE HAVEDB/DS = τNWHERE AT ANY POINT P[X,Y,Z] ON THE CURVEB IS UNIT VECTOR CALLED BINORMAL TO THE…

Q: Prove that all normal to the curve x=acos+atsint,y=asint-atcost. Are at a distance a from the…

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Q: 21. r(t) = t³ j + t² k %3D

A: Given  r(t)=t3j+t2k

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

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Q: Suppose that the plane tangent to the surface z = f(x, y) at (-4, 2, 5) has equation z + 5x + 4y =…

A: From the equation of tangent, we can derive: z = -(7 + 5x + 4y) Hence,

Q: |x'y" -y': y'a - y'= es K(t) = (x2 + у'?

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Q: Exercise 10. Show that the curvature of a planar curve which satisfies the equation y = f(r) is…

A: To Show- The curvature of a planar curve, which satisfies the equation y=fx is given by…

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem states that "for closed curve C enclosed a region R then the line integral along the…

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Q: b.) show that brnormal of the muolute the unit *-ズ+(c-s)で g =dG) |k-s) K| K* where K is the…

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Q: find the points on the surface + z² = 1 that have their tangent plane perpendicular to the vector…

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Q: Show that the curvature of a circle of radius r is K =1/ r.

A: The equation of the circle with radius r is y=±r2-x2. The upper portion of half circle equation is…

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Q: 2. Show that if a : [a, b] → R" is a regular parameterization of a curve then the curvature at α(t)…

A: Let,  is a regular parameterization of a curve To show the curvature at  is

Q: By using Green's theorem evaluate £ (Ary – y) dr + (** + 3ry) dy where C is a closed curve starting…

A: Green's theorem in a plane: If D is a region inclosed by a closed curve C then ∮C u dx+v dy=∫∫D…

Q: Consider 2-dimensional hyperbolic space with metric ds = dr'+dy and y > 0. The length of a line…

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Q: 10.) At what point does the curve have maximum Curvature? y=In(x). K(x)= f"|

A: firstly find f, f' and f'' . And substitute all these values in given curvature formula.  After that…

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Q: Determine the maximum curvature for the graph of f(x) = 9 In (8x). The maximum curvature is at x=

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Q: Let C be a curve given by the intersection of the surfaces z = x2 +y2;z = 3−2x . The value of the…

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Q: Let r (t,t,t) be a curve in 3-dimensional space having unit tangent and norma V2 vectors u and n.…

A: We will use formula of curvature of a curve to solve it.

Q: Suppose r(t) = i+t²j+2tk. Find the length of the portion of this curve that goes from the point G,…

A: We will find out the required value of length and expression for curvature.

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

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Q: Let 3 : (3t, r = be a curve in 3-dimensional space having unit tangent and normal vectors u and .…

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Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

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Q: Show that the Gauas and mean curvatures on , X = (u + v,u –- v,uv) at u = 1, v = 1 are K = 116 and H…

A: Given, X=u+v,u-v,uv u=1,v=1 , k=116 and H=182 We know that, E=xu,xuF=xu,xvG=xv,xv So, we need to…

Q: 8) If curvature of a space curve at (2,3) is 8 and curvature at (7,1) t=7 is 1, what can be said…

A: Given that curvature of a space curve at 2,3 is 8 and curvature at 7,1 is 1.

Q: 4. Find the curvature of r(t) = (tv2, e', e-t) at the point (0,1,1).

A: